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Hadamard code

About: Hadamard code is a(n) research topic. Over the lifetime, 1049 publication(s) have been published within this topic receiving 12130 citation(s). The topic is also known as: Walsh code & Walsh–Hadamard code.

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MonographDOI: 10.1515/9781400842902
Abstract: In Hadamard Matrices and Their Applications, K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use. The first half of the book explains the state of our knowledge of Hadamard matrices and two important generalizations: matrices with group entries and multidimensional Hadamard arrays. It focuses on their applications in engineering and computer science, as signal transforms, spreading sequences, error-correcting codes, and cryptographic primitives. The book's second half presents the new results in cocyclic Hadamard matrices and their applications. Full expression of this theory has been realized only recently, in the Five-fold Constellation. This identifies cocyclic generalized Hadamard matrices with particular "stars" in four other areas of mathematics and engineering: group cohomology, incidence structures, combinatorics, and signal correlation. Pointing the way to possible new developments in a field ripe for further research, this book formulates and discusses ninety open questions.

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Topics: Complex Hadamard matrix (71%), Hadamard matrix (71%), Hadamard three-lines theorem (67%) ...read more

492 Citations


Open access
Jennifer Seberry1, Mieko Yamada1Institutions (1)
01 Jan 1992-
Abstract: One hundred years ago, in 1893, Jacques Hadamard [31] found square matrices of orders 12 and 20, with entries ±1, which had all their rows (and columns) pairwise orthogonal. These matrices, X = (Xij), satisfied the equality of the following inequality, |detX|2 ≤ ∏ ∑ |xij|2, and so had maximal determinant among matrices with entries ±1. Hadamard actually asked the question of finding the maximal determinant of matrices with entries on the unit disc, but his name has become associated with the question concerning real matrices. Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry and Mieko Yamada, Hadamard matrices, Sequences, and Block Designs, Contemporary Design Theory – A Collection of Surveys, (D. J. Stinson and J. Dinitz, Eds.)), John Wiley and Sons, (1992), 431-560. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1070 11 "-. Hadamard Matrices, Sequences, and Block Designs Jennifer Seberry and Mieko Yamada 1 IN1RODUCTION 2 HADAMARD MATRICES 3 THE SmONGEST HADAMARD CONSmUCTION THEOREMS 4 ORTIIOGONAL DESIGNS AND AsYMPTOTIC EXISTENCE 5 SEQUENCES 6 AMICABLE HADAMARD MAmICES AND AOD 7 CoNSmUCTIONS FOR SKEW HADAMARD MAmICES 8 M -SmucTUREs 9 WILLIAMSON AND WILUAMSON-TYPE MAmICES 10 SBIBD AND THE EXCESS OF HADAMARD MATRICES 11 CoMPLEX HADAMARD MATRICES APPENDIX REFERENCES

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Topics: Hadamard transform (62%), Hadamard code (61%)

303 Citations


Open accessJournal ArticleDOI: 10.1109/LPT.2004.833859
Abstract: A new code structure for spectral-amplitude-coding optical code-division multiple-access system based on double-weight (DW) code families is proposed. The DW code has a fixed weight of two. By using a mapping technique, codes that have a larger number of weights can be developed. Modified double-weight (MDW) code is a DW code family variation that has variable weights of greater than two. The newly proposed code possesses ideal cross-correlation properties and exists for every natural number n. Based on theoretical analysis and simulation, MDW code is shown here to provide a much better performance compared to Hadamard and modified frequency-hopping codes.

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Topics: Constant-weight code (72%), Systematic code (69%), Polynomial code (69%) ...read more

274 Citations


Open accessJournal ArticleDOI: 10.1007/S11080-006-8220-2
Abstract: Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2,..., 16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12,14 and 16, which we could not find in the existing literature.

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Topics: Complex Hadamard matrix (79%), Hadamard matrix (73%), Hadamard three-lines theorem (71%) ...read more

273 Citations


Open accessJournal ArticleDOI: 10.1214/AOS/1176344370
Abstract: An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.

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Topics: Hadamard matrix (77%), Hadamard transform (67%), Hadamard code (64%) ...read more

272 Citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202112
202011
201927
201817
201719
201633

Top Attributes

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Topic's top 5 most impactful authors

Christos Koukouvinos

9 papers, 120 citations

Hilal A. Fadhil

5 papers, 119 citations

Hakob Sarukhanyan

5 papers, 12 citations

Ali N. Akansu

5 papers, 19 citations

Martin Bossert

4 papers, 29 citations

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